Question: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $16.7$ years; the standard deviation is $3.9$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living less than $12.8$ years.
Solution: $16.7$ $12.8$ $20.6$ $8.9$ $24.5$ $5$ $28.4$ $68\%$ $16\%$ $16\%$ We know the lifespans are normally distributed with an average lifespan of $16.7$ years. We know the standard deviation is $3.9$ years, so one standard deviation below the mean is $12.8$ years and one standard deviation above the mean is $20.6$ years. Two standard deviations below the mean is $8.9$ years and two standard deviations above the mean is $24.5$ years. Three standard deviations below the mean is $5$ years and three standard deviations above the mean is $28.4$ years. We are interested in the probability of a seal living less than $12.8$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $68\%$ of the seals will have lifespans within 1 standard deviation of the average lifespan. The remaining $32\%$ of the seals will have lifespans that fall outside the shaded area. Because the normal distribution is symmetrical, half $({16\%})$ will live less than $12.8$ years and the other half $({16\%})$ will live longer than $20.6$ years. The probability of a particular seal living less than $12.8$ years is ${16\%}$.